Finite Group Databasebeta

← #10950948

Subgroups — #10950948

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic, complemented
2#83742C211yesmaximal, characteristic
3C2×C3×C22⋊C16C211yesmaximal, characteristic, complemented
4#71416C212yesmaximal, complemented
5#71414C212yesmaximal, complemented
6#69113C211yesmaximal, characteristic, complemented
7#69112C211yesmaximal, characteristic, complemented
8#69111C212yesmaximal, complemented
9#69100C212yesmaximal, complemented
10#69098C211yesmaximal, characteristic, complemented
11#69095C211yesmaximal, characteristic, complemented
12#33756—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14#5791—21no—
15C2×#924—21no—
16C22×Dic12C2211yescharacteristic
17C2×#904C2211yescharacteristic
18C22×C3×C16C2212yes—
19#5116—21no—
20#5116—21no—
21#5116C2211yescharacteristic
22#5116C2211yescharacteristic
23#5116C2212yes—
24#5116C2212yes—
25#5115C2211yescharacteristic
26#5115C2211yescharacteristic
27#5114—21no—
28#5114—21no—
29#5113—21no—
30#5106C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#816—21no—
33C2×Dic24C2212yescomplemented
34C2×#800C2212yescomplemented
35C3×C22⋊C16C2211yescharacteristic
36C3×C22⋊C16C2211yescharacteristic
37C3×C22⋊C16C2212yes—
38#4509C2212yescomplemented
39#4505C2212yescomplemented
40#4504C2212yescomplemented
41#4503C2212yescomplemented
42#4503C2212yescomplemented
43#4503C2212yescomplemented
44#3050—31no—
45C2×#467—31no—
46#2237—32no—
47#2235—31no—
48#2232—31no—
49#2217—32no—
50#2216—32no—
51#2189—32no—
52#2188—31no—
53#2185—31no—
54C2×C22⋊C16D311yescharacteristic, complemented
55C23×C3×C4D411yescharacteristic
56C2×#235—21no—
57C22×Dic6—21no—
58C22×C3×C8—42no—
59C22×C3×C8D411yescharacteristic
60C22×C3×C8D411yescharacteristic
61C22×C3×C8D411yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65#953—41no—
66C2×#217—41no—
67#930—21no—
68#930—21no—
69#929—41no—
70C4×#67—41no—
71C4×#67—41no—
72#926—21no—
73C2×C4⋊Dic3—21no—
74#924—22no—
75#924—22no—
76#924—21no—
77#924—21no—
78#924—41no—
79#924—41no—
80C2×Dic3⋊C4—41no—
81C2×Dic12—21no—
82C2×Dic12—42no—
83C2×Dic12C2312yes—
84C2×Dic12C2311yescharacteristic
85#904—42no—
86C2×D24C2311yescharacteristic
87C2×#204—21no—
88C2×C3×C16—42no—
89C2×C3×C16C2312yes—
90C2×C3×C16C2312yes—
91#821—21no—
92#821—21no—
93#818—21no—
94#818C2311yescharacteristic
95#818C2311yescharacteristic
96#818C2312yes—
97#817—21no—
98#816—22no—
99#816—22no—
100#816—21no—
101#816—21no—
102Dic24—42no—
103#800—42no—
104C2×#135—31no—
105C22×Q16—31no—
106C23×C8C2×D311yescharacteristic
107#525—61no—
108C2×C22⋊Q8—61no—
109C2×Q32—32no—
110C2×SD32—32no—
111C22×C16C2×D312yes—
112#467—31no—
113#467—32no—
114#467—31no—
115#467—32no—
116#467—61no—
117#467—61no—
118#466—31no—
119#466—31no—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107—31no—
124#416—61no—
125C2×#103—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#358—32no—
130#358—32no—
131#357—32no—
132C22⋊C16C2×D311yescharacteristic
133C22⋊C16C2×D311yescharacteristic
134C22⋊C16C2×D312yes—
135C24×C3D811yescharacteristic
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4D811yescharacteristic
139C22×C3×C4D811yescharacteristic
140C22×C3×C4D811yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149#235—42no—
150#235—42no—
151C2×D12—21no—
152C2×C4×D3—41no—
153C2×Dic6—21no—
154C2×Dic6—22no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6—41no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—41no—
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2411yescharacteristic
168#217—42no—
169#217—42no—
170#217—41no—
171#217—41no—
172#217—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175C4⋊Dic3—22no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—41no—
179Dic3⋊C4—42no—
180Dic3⋊C4—42no—
181Dic3⋊C4—41no—
182Dic3⋊C4—41no—
183C4×Dic3—41no—
184Dic12—22no—
185Dic12—22no—
186Dic12—22no—
187Dic12—42no—
188D24—22no—
189#204—42no—
190C3×C16—22no—
191C3×C16—22no—
192C2×#41—61no—
193C22×Q8—61no—
194C23×C4D1211yescharacteristic
195#135—122no—
196C2×Q16—32no—
197C2×Q16—31no—
198C2×Q16—61no—
199C2×Q16—122no—
200C2×SD16—61no—
201C2×D8—31no—
202C22×C8—42no—
203C22×C8D1211yescharacteristic
204C22×C8D1211yescharacteristic
205C22×C8D1211yescharacteristic
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209#123—121no—
210C22⋊Q8—62no—
211C22⋊Q8—62no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C22⋊Q8—121no—
215C22⋊Q8—121no—
216#121—61no—
217#121—61no—
218#120—121no—
219C4×D4—121no—
220C4×D4—121no—
221#117—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225Q32—122no—
226SD32—122no—
227C2×C16—42no—
228C2×C16C22×D312yes—
229C2×C16C22×D312yes—
230#107—32no—
231#107—31no—
232#107—31no—
233#107—61no—
234#106—61no—
235#103—62no—
236#103—62no—
237#103—61no—
238#103—61no—
239#102—61no—
240#102—61no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3C2×D811yescharacteristic
245C23×C3C2×D811yescharacteristic
246C23×C3C2×D811yescharacteristic
247C22×D3—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4C2×D811yescharacteristic
258C2×C3×C4C2×D811yescharacteristic
259C2×C3×C4C2×D811yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261#67—82no—
262#67—82no—
263#67—82no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—82no—
270C2×Dic3—82no—
271D12—41no—
272D12—41no—
273C4×D3—82no—
274Dic6—41no—
275Dic6—42no—
276Dic6—42no—
277Dic6—41no—
278Dic6—82no—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—21no—
282C3×C8H(5,𝔽2)12yes—
283C24D2411yescharacteristic
284#41—122no—
285#41—122no—
286C2×Q8—61no—
287C2×Q8—62no—
288C2×Q8—122no—
289C2×Q8—122no—
290C2×Q8—121no—
291C2×D4—61no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4D2411yescharacteristic
301C22×C4D2411yescharacteristic
302C22×C4D2411yescharacteristic
303C22×C4C2×D1211yescharacteristic
304C22×C4C2×D1211yescharacteristic
305C22×C4C2×D1211yescharacteristic
306Q16—62no—
307Q16—62no—
308Q16—62no—
309Q16—122no—
310SD16—122no—
311D8—62no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C2×D1211yescharacteristic
319C2×C8C2×D1211yescharacteristic
320C2×C8C2×D1211yescharacteristic
321C2×C8C23×D311yescharacteristic
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—62no—
325C4⋊C4—121no—
326C4⋊C4—122no—
327C4⋊C4—122no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—122no—
331C22⋊C4—122no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—121no—
338C16—22no—
339C16—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3#50812yes—
349C22×C3#51012yes—
350C22×C3C22×D811yescharacteristic
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4#53411yescharacteristic
358C3×C4#57511yescharacteristic
359Dic3—82no—
360Dic3—81no—
361Dic3—81no—
362Dic3—82no—
363Dic3—81no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—121no—
368C23C2×D2411yescharacteristic
369C23C2×D2411yescharacteristic
370C23C2×D2411yescharacteristic
371Q8—122no—
372Q8—121no—
373Q8—121no—
374Q8—122no—
375Q8—242no—
376D4—121no—
377D4—121no—
378D4—242no—
379D4—242no—
380D4—242no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—242no—
396C2×C4—242no—
397C2×C4—242no—
398C2×C4C2×D2411yescharacteristic
399C2×C4C2×D2411yescharacteristic
400C2×C4C2×D2411yescharacteristic
401C2×C4C22×D1211yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8#100412yes—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3#318611yescharacteristic
410C2×C3C2×#50811yescharacteristic
411C2×C3C2×#51011yescharacteristic
412D3—161no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22#490412yes—
425C22#490912yes—
426C22C22×D2411yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—241no—
431C4—242no—
432C4—242no—
433C4—241no—
434C4—241no—
435C4#549311yescharacteristic
436C4#574811yescharacteristic
437C3#3375611yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—481no—
442C2#7861111yescharacteristic
443C2C2×#490411yescharacteristic
444C2C2×#490911yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×C22⋊C16C2
socleC22×C3C22×D8
prime radical#83742C2
hypercenterC2×C22⋊C16D3
hypocenterC3#33756
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#5116C2semidirect
3. C2×C3×C22⋊C16C2semidirect
4. #71416C2semidirect
5. #71414C2semidirect
6. #69113C2semidirect
7. #69112C2semidirect
8. #69111C2semidirect
9. #69100C2semidirect
10. #69098C2semidirect
11. #69095C2semidirect
33. C2×Dic24C22semidirect
34. C2×#800C22semidirect
38. #4509C22semidirect
39. #4505C22semidirect
40. #4504C22semidirect
41. #4503C22semidirect
42. #4503C22semidirect
43. #4503C22semidirect
54. C2×C22⋊C16D3semidirect
437. C3#33756semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#51162
#837422
C2×C3×C22⋊C162
#714162
#714142
#691132
#691122
#691112
#691002
#690982
#690952
#337563